How to solve this logarithmic equation $\ln(x/C_1)+C_2/x=C_1/C_2$

Question

I have this equation which represents two intersection points of a graph with a line. Can anyone help me please to solve it for $x$?

$C_1$, $C_2$ are constants. $x$ is a variable. The first solution is $x=C_1$, but I am looking for the other one please.

$$ \ln⁡(\frac{x}{C_1} )+ \frac{C_2}{x}=\frac{C_2}{C_1} $$ Many thanks.

Answer 1

\begin{align} \ln{\left(\frac{x}{C_1}\right)}+\frac{C_2}{x}=\frac{C_1}{C_2} &\iff\frac{x}{C_1}\exp{\left(\frac{C_2}{x}\right)}=\exp{\left(\frac{C_1}{C_2}\right)}\\ &\iff\frac{C_1}{x}\exp{\left(-\frac{C_2}{x}\right)}=\exp{\left(-\frac{C_1}{C_2}\right)}\\ &\iff-\frac{C_2}{x}\exp{\left(-\frac{C_2}{x}\right)}=-\frac{C_2}{C_1}\exp{\left(-\frac{C_1}{C_2}\right)}\\ &\iff-\frac{C_2}{x}=W_k\left(-\frac{C_2}{C_1}\exp{\left(-\frac{C_1}{C_2}\right)}\right)\\ &\iff x=-\frac{C_2}{W_k\left(-\frac{C_2}{C_1}\exp{\left(-\frac{C_1}{C_2}\right)}\right)}\\ \end{align} where $W_k(z)$ denotes the $k$th branch of the Lambert-W function. Here I am assuming the principal branch of the complex natural logarithm.